Backward bifurcation of an epidemic model with treatment. (English) Zbl 1093.92054

Summary: An epidemic model with a limited resource for treatment is proposed to understand the effect of the capacity for treatment. It is assumed that the treatment rate is proportional to the number of infectives below the capacity and is a constant when the number of infectives is greater than the capacity. It is found that a backward bifurcation occurs if the capacity is small. It is also found that there exist bistable endemic equilibria if the capacity is low.


92D30 Epidemiology
34C60 Qualitative investigation and simulation of ordinary differential equation models
34D23 Global stability of solutions to ordinary differential equations
Full Text: DOI


[1] Arino, J.; McCluskey, C.C.; van den Driessche, P., Global results for an epidemic model with vaccination that exhibits backward bifurcation, SIAM J. appl. math., 64, 260, (2003) · Zbl 1034.92025
[2] Anderson, R.M.; May, R.M., Infectious diseases of humans, dynamics and control, (1991), Oxford University Oxford
[3] Brauer, F.; Castillo-Chavez, C., Mathematical models in population biology and epidemiology, Texts in applied mathematics, 40, (2001), Springer New York · Zbl 0967.92015
[4] Britton, N.F., Essential mathematical biology, (2003), Springer London · Zbl 1037.92001
[5] Diekmann, O.; Heesterbeek, J.A.P., Mathematical epidemiology of infectious diseases. model building, analysis and interpretation, Wiley series in mathematical and computational biology, (2000), John Wiley and Sons Chichester · Zbl 0997.92505
[6] Dushoff, J.; Huang, W.; Castillo-Chavez, C., Backwards bifurcations and catastrophe in simple models of fatal diseases, J. math. biol., 36, 227, (1998) · Zbl 0917.92022
[7] Feng, Z.; Thieme, H.R., Recurrent outbreaks of childhood diseases revisited: the impact of isolation, Math. biosci., 128, 93, (1995) · Zbl 0833.92017
[8] Hadeler, K.P.; van den Driessche, P., Backward bifurcation in epidemic control, Math. biosci., 146, 15, (1997) · Zbl 0904.92031
[9] Hethcote, H.W., The mathematics of infectious disease, SIAM rev., 42, 599, (2000) · Zbl 0993.92033
[10] Hyman, J.M.; Li, J., Modeling the effectiveness of isolation strategies in preventing STD epidemics, SIAM J. appl. math., 58, 912, (1998) · Zbl 0905.92027
[11] Inaba, H.; Sekine, H., A mathematical model for chagas disease with infection-age-dependent infectivity, Math. biosci., 190, 39, (2004) · Zbl 1049.92033
[12] Liu, W.M.; Levin, S.A.; Iwasa, Y., Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. math. biol., 23, 187, (1986) · Zbl 0582.92023
[13] Lizana, M.; Rivero, J., Multiparametric bifurcations for a model in epidemiology, J. math. biol., 35, 21, (1996) · Zbl 0868.92024
[14] Martcheva, M.; Thieme, H.R., Progression age enhanced backward bifurcation in an epidemic model with super-infection, J. math. biol., 46, 385, (2003) · Zbl 1097.92046
[15] Ruan, S.; Wang, W., Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. differ. equat., 188, 135, (2003) · Zbl 1028.34046
[16] van den Driessche, P.; Watmough, J., A simple SIS epidemic model with a backward bifurcation, J. math. biol., 40, 525, (2000) · Zbl 0961.92029
[17] Takeuchi, Y.; Ma, Wanbiao; Beretta, E., Global asymptotic properties of a delay SIR epidemic model with finite incubation times, Nonlinear anal., 42, 931, (2000) · Zbl 0967.34070
[18] Wang, W.; Ma, Z., Global dynamics of an epidemic model with time delay, Nonlinear anal. real world appl., 3, 365, (2002) · Zbl 0998.92038
[19] Wang, W.; Ruan, S., Bifurcation in an epidemic model with constant removal rate of the infectives, J. math. anal. appl., 291, 775, (2004) · Zbl 1054.34071
[20] Wu, L.; Feng, Z., Homoclinic bifurcation in an SIQR model for childhood diseases, J. differ. equat., 168, 150, (2000) · Zbl 0969.34042
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.